Divide using synthetic division $$ ( 2x^{2}+18x+4 ) \div ( -2x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&2&18&4\\& & 2& \color{black}{20} \\ \hline &\color{blue}{2}&\color{blue}{20}&\color{orangered}{24} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+18x+4 }{ x-1 } = \color{blue}{2x+20} ~+~ \frac{ \color{red}{ 24 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&18&4\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ 2 }&18&4\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 2 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&18&4\\& & \color{blue}{2} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ 2 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrr}1&2&\color{orangered}{ 18 }&4\\& & \color{orangered}{2} & \\ \hline &2&\color{orangered}{20}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 20 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&2&18&4\\& & 2& \color{blue}{20} \\ \hline &2&\color{blue}{20}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 20 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrr}1&2&18&\color{orangered}{ 4 }\\& & 2& \color{orangered}{20} \\ \hline &\color{blue}{2}&\color{blue}{20}&\color{orangered}{24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x+20 } $ with a remainder of $ \color{red}{ 24 } $.